Advertisements
Advertisements
प्रश्न
Find the sine of the angle between the vectors `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.
Advertisements
उत्तर
Given that `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.
We know that `|vec"a" xx vec"b"| = |vec"a"||vec"b"| sin theta`
∴ `vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(3, 1, 2),(2, -2, 4)|`
= `hat"i"(4 + 4) - hat"j"(12 - 4) + hat"k"(-6 - 2)`
= `8hat"i" - 8hat"j" - 8hat"k"`
`|vec"a" xx vec"b"| = sqrt((8)^2 + (-8)^2 + (-8)^2)`
= `sqrt(64 + 64 + 64)`
= `sqrt(192)`
= `sqrt(64 xx 3)`
= `8sqrt(3)`
`|vec"a"| = sqrt((3)^2 + (1)^2 + (2)^2)`
= `sqrt(9 + 1 + 4)`
= `sqrt(14)`
`|vec"b"| = sqrt((2)^2 + (-2)^2 + (4)^2)`
= `sqrt(4 + 4 + 16)`
= `sqrt(24)`
= `2sqrt(6)`
∴ `sin theta = |vec"a" xx vec"b"|/(|vec"a"||vec"b"|)`
= `(8sqrt(3))/(sqrt(14) * 2sqrt(6))`
⇒ `(4sqrt(3))/sqrt(84) = (4sqrt(3))/(2sqrt(21))`
= `2/sqrt(7)`
Hence, `sin theta = 2/sqrt(7)`.
APPEARS IN
संबंधित प्रश्न
If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.
Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel
If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.
Classify the following measures as scalar and vector.
10 kg
Two collinear vectors are always equal in magnitude.
Two collinear vectors having the same magnitude are equal.
Find the direction cosines of the vector `hati + 2hatj + 3hatk`.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.
Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.
Find a vector of magnitude 4 units which is parallel to the vector \[\sqrt{3} \hat{i} + \hat{j}\]
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (−6, 3), B (−2, −5)
Find \[\left| \vec{A} B \right|\] in each case.
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}\]
Find a unit vector parallel to the vector \[\hat{i} + \sqrt{3} \hat{j}\]
Find the angle between the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]
Find the angle between the vectors \[\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}\]
The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.
Dot products of a vector with vectors \[\hat{i} - \hat{j} + \hat{k} , 2\hat{ i} + \hat{j} - 3\hat{k} \text{ and } \text{i} + \hat{j} + \hat{k}\] are respectively 4, 0 and 2. Find the vector.
Show that the vectors \[\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3\hat{i} - 6 {j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 {k} \right)\] are mutually perpendicular unit vectors.
If \[\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}\] \[\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\] \[\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}\] find λ such that \[\vec{a}\] is perpendicular to \[\lambda \vec{b} + \vec{c}\]
Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1).
Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.
Show that the points \[A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] are the vertices of a right angled triangle.
Position vector of a point P is a vector whose initial point is origin.
If `veca, vecb, vecc` are vectors such that `[veca, vecb, vecc]` = 4, then `[veca xx vecb, vecb xx vecc, vecc xx veca]` =
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, internally the ratio 2:1.
